Oberwolfach Seminars

Conformal Differential Geometry

Q-Curvature and Conformal Holonomy

Authors: Baum, Helga, Juhl, Andreas

  • Reviews very recent developments
  • Compact introduction into an active field of research
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About this Textbook

Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy groups etc.) are of central significance in differential geometry and physics. Well-known examples of conformally covariant operators are the Yamabe, the Paneitz, the Dirac and the twistor operator. These operators are intimely connected with the notion of Branson’s Q-curvature. The aim of these lectures is to present the basic ideas and some of the recent developments around Q -curvature and conformal holonomy.

The part on Q -curvature starts with a discussion of its origins and its relevance in geometry and spectral theory. The following lectures describe the fundamental relation between Q -curvature and scattering theory on asymptotically hyperbolic manifolds. Building on this, they introduce the recent concept of Q -curvature polynomials and use these to reveal the recursive structure of Q -curvatures.

The part on conformal holonomy starts with an introduction to Cartan connections and its holonomy groups. Then we define holonomy groups of conformal manifolds, discuss its relation to Einstein metrics and recent classification results in Riemannian and Lorentzian signature. In particular, we explain the connection between conformal holonomy and conformal Killing forms and spinors, and describe Fefferman metrics in CR geometry as Lorentzian manifold with conformal holonomy SU(1,m).

Reviews

From the reviews:

“This book grew out of an Oberwolfach student seminar on recent developments in conformal differential geometry which took place in 2007. It splits into two chapters, which to a large extent are independent of each other. Each of the chapters is an extended version of a series of lectures presented by one of the authors during the seminar and offers a nice and easily readable survey of an active area of research in conformal differential geometry.”­­­ (Andreas Cap, Mathematical Reviews, Issue 2011 d)

Table of contents (2 chapters)

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-3-7643-9909-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA
  • ISBN 978-3-7643-9908-5
  • Free shipping for individuals worldwide
  • Online orders shipping within 2-3 days.
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Bibliographic Information

Bibliographic Information
Book Title
Conformal Differential Geometry
Book Subtitle
Q-Curvature and Conformal Holonomy
Authors
Series Title
Oberwolfach Seminars
Series Volume
40
Copyright
2010
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Basel
eBook ISBN
978-3-7643-9909-2
DOI
10.1007/978-3-7643-9909-2
Softcover ISBN
978-3-7643-9908-5
Series ISSN
1661-237X
Edition Number
1
Number of Pages
X, 152
Topics